Question

What are the zeros of the function f(x) = x4 − x2 − 2?

±3, ±i
±2, ±i
positive or negative square root of 2 comma ±i
positive or negative square root 3, ±i

Answer 1

To find the zeros of the function \( f(x) = x^4 - x^2 - 2 \), we can start by making a substitution to simplify the equation. Let \( y = x^2 \). Then, we can rewrite the function as:

\[
f(y) = y^2 - y - 2
\]

Now, we need to find the roots of this quadratic equation. We can factor it:

\[
y^2 - y - 2 = (y - 2)(y + 1) = 0
\]

Setting each factor to zero gives:

1. \( y - 2 = 0 \) → \( y = 2 \)
2. \( y + 1 = 0 \) → \( y = -1 \)

Now, we substitute back \( y = x^2 \):

1. For \( y = 2 \):
\[
x^2 = 2 \implies x = \pm \sqrt{2}
\]

2. For \( y = -1 \):
\[
x^2 = -1 \implies x = \pm i
\]

Thus, the zeros of the function \( f(x) = x^4 - x^2 - 2 \) are:

\[
\pm \sqrt{2}, \pm i
\]

So, the correct answer is:

**positive or negative square root of 2, ±i**.